The spectrum of the Chern subring

For certain subrings of the mod-p cohomology ring of a compact Lie group, we give a description of the prime ideal spectrum, analogous to Quillen's description of the spectrum of the whole ring. Examples of such subrings include the Chern subring (the subring generated by Chern classes of all unitary representations), and for finite groups the subring generated by Chern classes of representations realizable over any specified field. As a corollary, we deduce that the inclusion of the Chern subring in the cohomology ring is an F-isomorphism for a compact Lie group G if and only if the following condition holds: For any homomorphism f between elementary abelian p-subgroups of G such that f(v) is always conjugate to v, there is an element g of G such that f is equal to conjugation by g.